Answer :
To determine the end behavior of the given function [tex]\( f(x) \)[/tex] based on the table, let's analyze the values provided:
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -4 & 18 \\ \hline -3 & 9 \\ \hline -2 & 6 \\ \hline -1 & 3 \\ \hline 0 & 0 \\ \hline 1 & -3 \\ \hline 2 & -6 \\ \hline 3 & -9 \\ \hline 4 & -18 \\ \hline \end{array} \][/tex]
1. Analyzing [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] increases:
- When [tex]\( x \)[/tex] increases from 1 to 4, [tex]\( f(x) \)[/tex] decreases: [tex]\( -3, -6, -9, -18 \)[/tex].
- This trend suggests that as [tex]\( x \rightarrow \infty \)[/tex], [tex]\( f(x) \)[/tex] continues to decrease.
2. Analyzing [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] decreases:
- When [tex]\( x \)[/tex] decreases from -1 to -4, [tex]\( f(x) \)[/tex] decreases: [tex]\( 3, 6, 9, 18 \)[/tex].
- However, since these are positive values that eventually decrease (as observed from the entire trend from positive to negative values), we see that the function is decreasing overall.
Combining these observations:
- As [tex]\( x \rightarrow \infty \)[/tex], [tex]\( f(x) \rightarrow -\infty \)[/tex].
- As [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( f(x) \rightarrow -\infty \)[/tex].
Therefore, the best prediction for the end behavior of the graph of [tex]\( f(x) \)[/tex] is:
"As [tex]\( x \rightarrow \infty, f(x) \rightarrow -\infty \)[/tex], and as [tex]\( x \rightarrow -\infty, f(x) \rightarrow -\infty \)[/tex]."
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -4 & 18 \\ \hline -3 & 9 \\ \hline -2 & 6 \\ \hline -1 & 3 \\ \hline 0 & 0 \\ \hline 1 & -3 \\ \hline 2 & -6 \\ \hline 3 & -9 \\ \hline 4 & -18 \\ \hline \end{array} \][/tex]
1. Analyzing [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] increases:
- When [tex]\( x \)[/tex] increases from 1 to 4, [tex]\( f(x) \)[/tex] decreases: [tex]\( -3, -6, -9, -18 \)[/tex].
- This trend suggests that as [tex]\( x \rightarrow \infty \)[/tex], [tex]\( f(x) \)[/tex] continues to decrease.
2. Analyzing [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] decreases:
- When [tex]\( x \)[/tex] decreases from -1 to -4, [tex]\( f(x) \)[/tex] decreases: [tex]\( 3, 6, 9, 18 \)[/tex].
- However, since these are positive values that eventually decrease (as observed from the entire trend from positive to negative values), we see that the function is decreasing overall.
Combining these observations:
- As [tex]\( x \rightarrow \infty \)[/tex], [tex]\( f(x) \rightarrow -\infty \)[/tex].
- As [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( f(x) \rightarrow -\infty \)[/tex].
Therefore, the best prediction for the end behavior of the graph of [tex]\( f(x) \)[/tex] is:
"As [tex]\( x \rightarrow \infty, f(x) \rightarrow -\infty \)[/tex], and as [tex]\( x \rightarrow -\infty, f(x) \rightarrow -\infty \)[/tex]."
